\(S^3\) as a cover of \(S^3\) branched over a knot
A branched covering over a knot \(k\) is a covering space of the 3-sphere \(S^3\) whose branching set is the knot \(k\). \textit{R. Fox} studied [Proc. Univ. Georgia Inst. 213-216 (1961)] a certain class of irregular coverings of \(S^3\) branched along some knot or link which turned out to be homotopy spheres, and asked if these are \(S^3\). Then \textit{G. Burde} [Can. J. Math. 23, 84-89 (1971; Zbl 0205.53301)] and \textit{J. M. Montesinos} [Ph. D. Thesis, Madrid, 1971] showed that all the homotopy spheres constructed by Fox are \(S^3\). In this context, Montesinos asked in relation to Fox's question: ``Given a knot \(k\) in \(S^3\) is the 3-sphere a covering of \(S^3\) branched over \(k\)?''. In this paper the author answers Montesinos's question in the affirmative for torus knots and a certain class of satellites of torus knots which includes all iterated torus knots. The main results: Theorem 2.1. Let \(\tau_{p,q}\) be the \((p,q)\)-torus knot. Then, for every \(m\in\mathbb{N}\), there is an \(mpq\)-fold covering \(f_1:L(m,1) \to(S^3, \tau_{p,q})\) branched along \(\tau_{p, q}\) such that \(S_{f_1}\) is a fiber and \(f_1|S_{f_1}: S_{f_1}\to \tau_{p, q}\) is a homeomorphism. (Here \(\tau_{p,q}\) is a regular fiber of the Seifert bundle \(S^3=(0,0; r/p,s/q)\), where \(-r/p-s/q =\pm 1/pq\); \(L(m,1)\) is the lens space considered as the Seifert bundle \((0,0; \pm m/1)\) and \(S_{f_1}\) denotes the singular set of \(f_1)\). Corollary 2.2. Let \(\tau_{p,q}\) be the \((p,q)\)-torus knot in \(S^3\). Then, for any \(m\in\mathbb{N}\), there is an \(m^2pq\)-fold covering \(f:S^3 \to(S^3, \tau_{p,q})\) branched along \(\tau_{p,q}\). Moreover, \(S_f\) is a trivial knot and \(f|S_f:S_f \to\tau_{p,q}\) is an \(m\)-fold covering. Theorem 2.3. Let \(f:M^3\to S^3\) be a covering branched along the knot \(k\) such that \(S_f\) is connnected and \(f|S_f\) is \(m\) to 1. Let \(J(k)\) be a monotone satellite of \(k\) whose winding number \(d\) divides \(m\). Then there is a covering \(g:M^3\to S^3\) branched over \(J(k)\) which coincides with \(f\) outside a tubular neighborhood of \(S_f\). Corollary 2.4. The 3-sphere is an \(m^2pq\)-fold cover of \(S^3\) branched over \(J(\tau_{p,q})\), where \(\tau_{p,q}\) is the \((p,q)\)-torus knot, and the winding number of \(J(\tau_{p,q})\) is a divisor \(d\) of \(m\). As an application it is also proved: Theorem 3.2. Let \(\tau_{p, q}\) be the \((p,q)\)-torus knot in \(S^3\). Then infinitely many doubles of \(\tau_{p,q}\) are universal.
scientific article; zbMATH DE number 1787540